Source Panel Method Pressure Distribution Calculator

What is the Source Panel Method?

The source panel method is a powerful numerical technique in aerodynamics used to calculate pressure distribution using source panel elements over the surface of a 2D body, such as an airfoil. This method models the flow as inviscid and irrotational (potential flow). The surface of the airfoil is approximated by a series of straight-line segments called “panels.” A “source” of a certain strength is placed on each panel. The strength of these sources is calculated to satisfy a key boundary condition: that there is no flow through the surface of the airfoil. Once the source strengths are known, the velocity of the air over each panel can be determined. From the velocity, we can then calculate the pressure distribution using Bernoulli’s equation, which gives us the pressure coefficient (Cp).

The Source Panel Method Formula and Explanation

The core of the method is to solve a system of linear algebraic equations. For N panels, we have N unknown source strengths (λj). The governing equation enforces the zero-normal-flow condition at the center of each panel (the collocation point).

The velocity normal to panel ‘i’ induced by the freestream and all source panels ‘j’ is set to zero:

Σ [ A_ij * λ_j ] = -V_∞ * cos(β_i)

Once the source strengths (λ) are found, the tangential velocity (V_ti) on each panel is calculated:

V_ti = V_∞ * sin(β_i) + Σ [ B_ij * λ_j ]

Finally, the pressure coefficient (Cp) for each panel is found:

Cp_i = 1 – (V_ti / V_∞)²

Variable Explanations

Variable	Meaning	Unit	Typical Range
V∞	Freestream Velocity	m/s or ft/s	10 – 100 m/s (subsonic)
α	Angle of Attack	Degrees	-5° to 15°
N	Number of Panels	Unitless	20 – 200
λj	Source Strength of panel j	m²/s	Varies
Cpi	Pressure Coefficient of panel i	Unitless	-4.0 to 1.0

Practical Examples

Example 1: Standard Conditions

Let’s take a typical subsonic flight condition to calculate pressure distribution using source panel method.

• Inputs: Freestream Velocity = 50 m/s, Angle of Attack = 4°, Number of Panels = 60.
• Results: This will produce a classic airfoil pressure distribution. You will see a large negative Cp value (high velocity, low pressure) near the leading edge on the upper surface, indicating where lift is being generated. The stagnation point at the leading edge will have a Cp of +1.0.

Example 2: Effect of Angle of Attack

Increasing the angle of attack significantly changes the pressure distribution.

• Inputs: Freestream Velocity = 50 m/s, Angle of Attack = 10°, Number of Panels = 60.
• Results: Compared to the first example, the negative Cp peak on the upper surface will become much stronger (more negative), and the positive pressure area on the lower surface will increase. This signifies a higher lift force. However, increasing the angle too much can lead to flow separation (stall), which this inviscid model cannot predict. For more on this, you could read about flow separation.

How to Use This Calculator

Using this tool to calculate pressure distribution using source panel logic is straightforward:

1. Enter Freestream Velocity: Input the speed of the aircraft or the fluid.
2. Select Units: Choose between meters per second (m/s) or feet per second (ft/s).
3. Set Angle of Attack: Enter the angle of the airfoil relative to the flow, in degrees. Small positive angles are typical for generating lift.
4. Define Number of Panels: Choose how many panels to use. More panels give a more accurate shape but take longer to compute. An even number is required.
5. Click Calculate: The tool will solve the equations and display the results. The chart shows the pressure coefficient (Cp) along the airfoil chord, and the table provides the raw data. A Cp of 1 indicates the stagnation point (zero velocity), while negative values indicate flow faster than the freestream.

Key Factors That Affect Pressure Distribution

• Airfoil Shape: The curvature and thickness of the airfoil are the primary drivers of the pressure distribution.
• Angle of Attack (α): As α increases, the pressure on the lower surface generally increases, and the pressure on the upper surface decreases, leading to more lift.
• Freestream Velocity (V_∞): While Cp itself is dimensionless, the actual pressure difference (and thus lift) is proportional to the square of the velocity.
• Fluid Density (ρ): Higher density fluid (e.g., at lower altitudes) results in greater pressure differences for the same velocity.
• Viscosity: This model ignores viscosity, but in reality, viscosity creates a thin boundary layer that can slightly alter the “effective” shape of the airfoil and eventually cause flow separation. Interested in real-world effects? Learn about boundary layer theory.
• Compressibility: At high speeds (approaching the speed of sound), air density changes, and the simple incompressible formulas used here are no longer accurate. One might need to use a Prandtl-Glauert correction.

Frequently Asked Questions (FAQ)

1. What does a pressure coefficient (Cp) of 1.0 mean?

A Cp of 1.0 represents the stagnation point, where the flow comes to a complete stop relative to the airfoil. This typically occurs at the leading edge.

2. Why are the Cp values on the top surface negative?

Negative Cp values indicate that the pressure is lower than the freestream pressure. This is because the flow accelerates as it travels over the curved upper surface of the airfoil, causing a drop in pressure according to Bernoulli’s principle. This low pressure region is what generates most of the lift.

3. What are the limitations of the source panel method?

The primary limitation is that it assumes inviscid (frictionless) and incompressible flow. It cannot model viscous effects like drag from skin friction or flow separation (stall), which occur at high angles of attack.

4. Why do I need to use an even number of panels?

This calculator uses a NACA 0012 symmetric airfoil and distributes half the panels on the upper surface and half on the lower surface. An even number ensures this symmetry is maintained.

5. How does the number of panels affect the result?

More panels provide a better geometric approximation of the airfoil’s smooth curve, leading to a more accurate pressure distribution. However, there are diminishing returns, and computation time increases with the square of the number of panels.

6. Can this calculator be used for supersonic flow?

No, the formulas used here are based on potential flow theory, which is only valid for subsonic, incompressible flow. Supersonic flow involves shock waves and requires different governing equations.

7. What airfoil is this calculator using?

This calculator specifically models a symmetric NACA 0012 airfoil, a very common and well-studied profile used for research and general aviation. For details on other airfoils, you can explore the airfoil database.

8. How is lift calculated from the pressure distribution?

Lift is the net force acting perpendicular to the freestream direction. It is found by integrating the pressure distribution over the entire surface of the airfoil, taking into account the angle of each panel. This tool focuses on how to calculate pressure distribution using source panel methods, a prerequisite for finding lift.